Concept preview

What you will be able to do

1. 01Move between position, velocity $\vec v=\langle x',y'\rangle$, and acceleration $\vec a=\langle x'',y''\rangle$ by differentiating or integrating componentwise (with initial conditions).
2. 02Compute the speed $|\vec v|=\sqrt{(x')^2+(y')^2}$ at an instant and the total distance travelled $\int_a^b|\vec v|\,dt$ over an interval, distinguishing total distance from net displacement.
3. 03Decide when a particle is speeding up or slowing down using the sign of $\vec v\cdot\vec a$ — speeding up when it is positive.

A particle moving in the plane is described by its position vector $\vec r(t)=\langle x(t),\,y(t)\rangle$. Differentiating once gives the velocity $\vec v(t)=\vec r\,'(t)=\langle x'(t),\,y'(t)\rangle$, an arrow pointing in the direction of motion whose length is the speed. Differentiating again gives the acceleration $\vec a(t)=\vec v\,'(t)=\langle x''(t),\,y''(t)\rangle$. The chain runs both ways: from position you differentiate down to velocity and acceleration; from acceleration you integrate up to velocity and position, using an initial condition at each step to fix the constant vector — exactly the skill from 9.5. Everything in a motion problem is one of these three vectors or a scalar squeezed out of them.